How do you use the tangent line approximation to approximate the value of #ln(1004)# ?

1 Answer
Oct 28, 2014

A formula for a tangent line approximation of a function f, also called linear approximation , is given by

#f(x)~~f(a)+f'(a)(x-a),#
which is a good approximation for #x# when it is close enough to #a#.

I'm not sure, but I think the question is about approximate the value #ln(1.004)#. Could you verify it please? Otherwise we will need to know an approximation to #ln(10).#

In this case, we have #f(x)=ln(x)#, #x=1.004# and #a=1#.

Since,
#f'(x)=1/x=>f'(1)=1# and #ln(1)=0#, we get

#ln(1.004)~~ln(1)+1*(1.004-1)=0.004.#